If `l_(1), m_(1), n_(1), l_(2), m_(2), n_(2) and l_(3), m_(3), n_(3)` are direction cosines of three mutuallyy perpendicular lines then, the value of `|(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3))|` is

A=[{:(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)):}] and B=[{:(p_(1),q_(1),r_(1)),(p_(2),q_(2),r_(2)),(p_(3),q_(3),r_(3)):}] Where p_(i), q_(i),r_(i) are the co-factors of the elements l_(i), m_(i), n_(i) for i=1,2,3 . If (l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)) and (l_(3),m_(3),n_(3)) are the direction cosines of three mutually perpendicular lines then (p_(1),q_(1), r_(1)),(p_(2),q_(2),r_(2)) and (p_(3),q_(),r_(3)) are

(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)) are the direction cosines of three mutually perpendicular lines. If the line, whose direction ratios are l_(1)+l_(2)+l_(3),m_(1)+m_(2)+m_(3),n_(1)+n_(2)+n_(3) , makes angle theta with any of these three lines, then cos theta is equal to

The direction cosines of a line equally inclined to three mutually perpendiclar lines having direction cosines as l_(1),m_(1),n_(1),l_(2),m_(2),n_(2) and l_(3), m_(3),n_(3) are

Let l_(i),m_(i),n_(i);i=1,2,3 be the direction cosines of three mutually perpendicular vectors in space.Show that AA=I_(3) where A=[[l_(2),m_(2),n_(2)l_(3),m_(3),n_(3)]]

Let l_(1),m_(1),n_(1);l_(2),m_(2),n_(2) and l_(3),m_(3),n_(3) be the direction cosines of three mutually perpendicular lines.Show that the direction ratios of the line which makes equal angles with each of them are (l_(1)+l_(2)+l_(3)),(m_(1)+m_(2)+m_(3)),(n_(1)+n_(2)+n_(3))

If l_(1), m_(1), n_(1) and l_(2),m_(2),n_(2) are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_(1)n_(2)-m_(2)n_(1),n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1) .